Two-Dimensional Fluid Foams at Equilibriumgraner.net/francois/publis/2dfoams_Wuppertal.pdf · 2014-10-01 · Two-Dimensional Fluid Foams at Equilibrium 191 Minimisation and equilibration

Two-Dimensional Fluid Foams at Equilibrium

Francois Graner

Spectrometrie Physique, BP 87, F-38402 St Martin d’Heres Cedex, France;CNRS UMR 5588 et Universite Grenoble Ie-mail: [email protected].

Abstract. This is an introduction to fluid foams for non-specialists. After presenting some ap-plications and modeling of foams in general, we focus on ideal two-dimensional fluid foams atequilibrium. We discuss : the interplay of topology, geometry and forces; the central role of pres-sure and energy; an analogy with 2D electrostatics; the role of disorder; the minimal perimeterproblem.

1 Introduction

There is a place where chemists, engineers, mathematicians and physicists routinelymeet: congresses about foams [26, 40, 36]. In fact, fluid foams stimulate truly interdis-ciplinary fundamental research with a wide range of industrial applications. They haveeven fascinated some artists. We will try to explain here why foams are at the crossroadsof different approaches, and why they are so attractive.

Section 2 presents their main properties and the corresponding applications. Section3 tries to catch the essence of what a foam is, and how far we should simplifiy itto model it. Section 4 then proceeds by presenting the position of the problem forphysicists and mathematicians. In the remaining of the paper, we mostly focus on ourown research, which concerns the statistical physics of two-dimensional (2D) foams atequilibrium. Section 5 tries to simplify the problem and reduce it to known results ofelectrostatics; Sects. 6 and 7 apply it to determinations of the foam’s energy and pressurefield, respectively; Sect. 8 addresses the physical aspects of the mathematical “minimalperimeter problem”. For 3D aspects, which we do not treat here, we refer to excellentexisting books: for the mathematical approach, see [20]; for the physical and historicalpoint of view see [19, 35].

The branch of mathematics which deals with the minimal perimeter problem in two ormore dimensions is achieving remarkable progresses [20, 17, 18]. Readers interested inthe mathematical aspects of foams, especially their specific interplay between topology,geometry and pressure, might read Sects. 4, 5, 6, and 8. These readers can skip Sects. 3and 7, which are more intended for foam physicists.

However, the point of view we want to present here is completely different. It is theapproach of statistical physics: to start from exact detailed properties and deduce averageglobal properties. We have tried to mention in the text which formulas are approximate:this is the case for Sects. 5.2, 5.3, 6.1, 6.2, 7.2, 8.2, 8.3.

The morphology of foams,i.e. the shape of the cellular domains, may be quantifiedby the techniques presented in the contributions to this volume by Arnset al.(1.2) and by

K.R. Mecke, D. Stoyan (Eds.): LNP 600, pp. 187–211, 2002.c© Springer-Verlag Berlin Heidelberg 2002

188 Franc¸ois Graner

Fig. 1. Example of fluid foam, here the surface of a soap froth. Picture courtesy of S. Courty.

Beisbartet al. (3.2) applying Minkowski functionals on spatially disordered materials.Orientational and tensorial features of patterns are characterised by methods describedby P. Soille (3.1) and Beisbartet al. (4.2) in this volume, also useful in the context offoams.

2 A Wide Range of Properties and Applications

“Fluid foams” consist ofgas bubbles separated by a continuous film of liquid(Fig. 1).Their applications usually combine simultaneously several of their properties, related totheir structure, mechanics, chemistry, and surface tension. For reviews, see for instance[19, 39, 3, 12, 13, 25, 34, 11, 14, 33].

2.1 Structure

Since the gas occupies most of the volume, foams have a low density, and a high specificsurface per unit mass. They can uniformly cover large surfaces. They help in fire-fighting:for oil, which they cover and insulate from air (while water would only fall down, andpossibly later boil explosively), and in confined media; foams are much less destructivethan water, and the air they contain allows endangered persons to breathe.

Since they can fill volumes with convoluted shapes, they are easy to mould, hencean intermediate step in most processes to prepare solid foams [36]: for cushion stuffing,building materials, bumpers, filters, insulators, catalysts, heat exchangers. Fluid foamsfill and can clean pipes or pores, using a small amount of mass; this is essential in nucleardecontamination or chemical pollution where it is necessary to reduce the amounts of

Two-Dimensional Fluid Foams at Equilibrium 189

cleaning residues. They considerably reduce the cost of all fiber treatments: brushinghair, protecting crops; but also waterproofing or dying textiles, being easier and cheaperto remove by drying than bulk water.

2.2 Mechanics

Fill a bottle with water, add some dishwashing product, shut it with your thumb, andshake it: the foam damps the shock and your thumb can barely feel it. Such efficientenergy dissipation has shock damping applications from demining to gun silencers.

Other police and military applications use foams for confinement: as anti-riots agents,or for preparation and separate storage of H-bombs sub-components. Fluid foams are alsouseful as temporary insulators: spermicides, thermal insulation of greenhouses duringthe night, protection of wounds against toxic chemicals, trap for dust in coal mines,prevention of beer oxydation during bottling, prevention of polluant dispersion in soils.

Resistance to pressure is crucial in several applications of foams to oil drilling: aspressurising fluid carrying sand to fracture the rock; to carry away forage residues inunder-pressurised regions; to eliminate water; or, by filling empty pores, to prevent vapor,water or gas from leaking through porous rocks.

Note that a foam has a shape, to which it returns after a small mechanical perturbation:it is elastic. Under higher stress, it deforms plastically; under high shear rate, it flows.This triple, elasto-plasto-viscous behaviour, although not specific to foams, is one of theirmost attractive features. For instance, in shaving or in airplane windshield de-icing, theycan be deposed in large amounts on vertical walls thanks to their elasticity, and are laterevacuated through the shear produced by the blade or the wind, respectively.

2.3 Chemistry

Materials with high affinity for water, especially aqueous solutions, are called “hy-drophilic”. “Hydrophobic” materials are not wet by water; for instance oil and waterdo not spontaneously mix. Mixing oil with water, so that rinsing can evacuate the oil,requires cleaning agents, usually “amphiphilic” molecules with a hydrophilic head and along oily chain made of carbons. Amphiphilic molecules are often also foaming agents,so that foams appear as by-products in cleaning. Neither the presence, nor the stabilityof the foam seems to improve the quality of cleaning [6]. In that case their presence is amarker of the dose injected, and their destruction marks the presence of impurities.

The amphiphilicity of the foams implies they have an affinity for most materials.Fine tuning their chemical composition leads to selective affinity. This is used in minesto separate ore and useless materials in huge flotation tanks. Foams also can separatecolorants by fractional distillation.

2.4 Model Systems

In fundamental research, foams are used as model systems. Since they minimise theirsurface energy by decreasing the area of the bubbles, they are examples of the mathemat-ical “minimum perimeter” problem which we discuss below, and of minimal surfaces


used in architecture. They are often compared with other cellular patterns : solid foams,biological epithelia or aggregates, ordered pattern like the retina or honeycombs, fracturepattern like in basalts or salt lakes. However, the most useful analogy is with cellularpatterns which do have a surface energy and minimise it: grain boundaries in crystals,magnetic domains. Since their characteristics are entirely measurable from images, with-out any hidden variables, foams are also models for more complex elasto-plasto-viscousmaterials.

2.5 Other

Although the above list of applications is long and could be even longer, foams are oftenassociated with an image of futility. This might originate from their consistence andshort life-expectancy. But there might be another reason: foams which are visible in ourdaily life have less essential applications; or often no applications at all, when they areby-products or polluants.

As light and attractive materials, they are essential to the image of foods and espe-cially drinks like sparkling wines, sodas, or beers, which they protect from oxydation.Without attempting at listing all representation of bubbles and foams in poetry or pictures[19, 9], let us mention “Soap bubbles”, by Edouard Manet, or a delicateazulejoin thenorth-east corner of the garden, in Palacio de la Frontera near Lisbon. They appear inmovies, in foam parties, or in both together: “The Party”, by Blake Edwards, ends withan unforgettable invasion by a foam provoked by Peter Sellers.

3 A Generic Model for Fluid Foams

3.1 Common Characteristics

Foams exist if and only if two generic conditions are met:(i) within each bubble, the matter is conserved;and:(ii) the energy decreases when the total surface of the walls decreases.This deserves some discussion and examples.

In a foam the surface energy is an increasing, but not necessarily a linear functionof the surface. For instance, the surface tension can be non-uniform due to temperaturegradient: Marangoni effect. Or, a crystal usually consists in many small monodomaincrystals; the boundaries between these grains actually meet both above conditions: herethe surface energy is not uniform nor anisotropic. To the contrary, condition (ii) excludesaggregates of vesicles, although they look like soap bubbles [27]: since their walls do notcommunicate together, each vesicle has a different tension; the total energy is a functionof each vesicle’s surface separately, not of their sum.

In addition to this surface energy, the total energy of the foam always containsadditional, specific terms which vary from foam to foam. For instance, the repulsionbetween the electrostatic dipoles of amphiphilic molecules on either side of the filmprovides a microscopic energy term to stabilise the walls against breakage.


Minimisation and equilibration of energy requires both phases to be fluid. Thisincludes emulsions, which are fluid-in-fluid foams, like oil in water. This specificallyexcludes solid foams, granular systems like dry or wet sand.

Due to condition (i), an equation of state relates the pressure and volume of eachbubble separately. The surface of the walls decrease and the pressure within each bubbleincreases until they exactly balance each other. This ensures the mechanical stability ofthe foam. Two simple cases are often studied: the perfect gas, where the pressure andvolume are inversely proportional; and the incompressible emulsion, where the volumeis constant and only the pressure varies.

To observe a foam, conditions (i) and (ii) need only be valid during the time requiredto reach the mechanical equilibrium: typically tens of milliseconds for a soap froth.Almost all real foams violate these conditions at longer time scales. A main cause iswall breakage: foams produced without amphiphilic molecules by violent mixing gasand liquid (froth on sea waves) or chemical reactions (metallic foams) are transient andout-of-equilibrium, and only last a few seconds. A slower phenomenon is drainage ofthe water under gravity, like in mine flotation tanks; in a beer glass it only takes secondsor minutes. Finally, depending on the solubility of the gas in the liquid, bubbles canlose gas into their neighbours and disappear, resulting in the foam losing its bubbles andcoarsening over minutes or hours.

3.2 The Simplest Foam

Ideal foam: To catch the essence of the physics common to all foams, we focus on aidealised foam defined as follows.• Dry foam: the gas occupies most of the foam’s volume, while the liquid occupies avery small fraction. In physical experiments this dryness is limited by the stability of thefilm: typically, in soap froths it can go down to one or two percents. In the ideal foamwe take the limit of fluid fraction going to zero.• Incompressible foam:Although foams are full of gas, we can usually neglect theircompressibility: of orderP−1

atm = 10−5 m3.J−1, that of a perfect gas. This is muchsmaller than the effect of the surface tension, characterised by the Laplace overpressure(2) of order of the bubble wall curvature, typically a centimeter, divided by the surfaceenergy, typically10−2J.m−2, hence1 m3.J−1. Emulsions, which are full of liquid, areof course even less compressible.• Energy proportional to the surface:this means that surface tension is uniform andanisotropic, and dominates all other contributions to energy. A soap froth with bubblesmuch larger than molecular sizes often provides a correct example.• No temperature effect:room temperature does not play any significant role in thestructure of a foam, see Sect. 8.1 for order of magnitudes.

Two-dimensional (2D) foams: We chose to focus on two dimensional (2D) foams,although specialists of actual (3D) experiments often object to this choice.• The first usual objection to 2D foams is their lack of practical applications by them-selves. The answer is that they turn particularly useful via their use in fundamental


research. 2D experiments and simulations are easier to realise and, what is more impor-tant, to analyse both qualitatively and quantitatively. Theories are also easier to developin 2D, before generalisation to 3D, especially since there are more exact results in 2D;this will be addressed in Sect. 4.• The second, more radical objection is that experimental foams are never truly 2D. Theanswer is that it does not really matter. To understand the physics, we perform truly 2Dsimulations and theories, and approximately 2D experiments. An experiment is a correctmodel of 2D foam if the energy is proportional to the total perimeter. For instance, in asoap froth between two parallel plates of glass, each line has an energy proportional to itslength, 3D effects such as the curvature in the vertical plane appearing only in the valueof the proportionality constant. Some 2D foams are stabilised by dipolar interactionswithout amphiphilic molecules [29]: in Langmuir monolayers [4], magnetic garnets [37],oil/ferrofluid emulsions [7]; they are acceptable model of 2D foams if and only if thedipolar energy is small enough not to affect their behaviour.

4 Position of the Problem

From now on, we consider only ideal, dry 2D foams at equilibrium. This section dealswith its structure: first the position of the problem, then some directions of research.Most results below come from [15], to which we refer for details.

4.1 Energy

An ideal 2D foam consists inN bubbles with fixed areas{Ai, i = 1, . . . , N}, Fig. 2.Its energy [41] is simply the sum of wall lengths:

H = γ∑

0≤i<j≤N

ij = λ

N∑i,j=0

ij = λ

N∑i=0

Li, (1)

whereij is the length of the wall between bubblesi andj, Li =∑

j ij is the perimeterof bubblei, i = 0 denotes the medium that surrounds the foam,ii = 0. Hereλ is theline energy of the interface between water and gas,γ = 2λ is the line energy of a wallconsisting of two interfaces, both in J.m−1.

If 〈A〉 is the average bubble area, note thatH/λ 〈A〉1/2 is a dimensionless functionof the bubbles’ shapes only. In this sense, minimising the energy is a purely geometricalproblem, universal in the sense that it is the same foranyperimeter-minimising system.

4.2 Local Rules: Plateau

At equilibrium, the ideal foam obeys the following local rules, stated by Plateau [24].


Fig. 2. Example of a 2D foam, here a soap froth between two plates of glass. The line tensionλis here twice the usual surface tension between water and gas (here 27.3 mN/m) multiplied by thedistance between plates (1 mm). Picture courtesy of O. Lordereau.

Walls: According to Laplace’s law, the wall between bubblesi, j has a curvature whichbalances the 2D pressure difference:

κij =Pi − Pj

γ. (2)

whereκij = −κji is the algebraic curvature, counted positive when bubblei is convexcompared with bubblej. As a consequence, along the wall, the curvature is uniform:each wall is an arc of circle.

Vertices: Bubble edges meet in triples at2π/3 angles (Fig. 3). Foams have the topologyof three-fold connected networks; four-fold vertices cost more energy and are unstable(Fig. 7).

As a consequence of (2), algebraic curvatures of the three edges that meet at thesame vertex must add to zero:

κij + κjk + κki =Pi − Pj

γ+Pj − Pk

γ+Pk − Pi

γ= 0. (3)

Equation (3) holds for any closed contour crossing more edges.


Fig. 3. Illustration of Plateau rules, on a 2D soap froth. Bubble walls are arcs of circles meetingby three at (approximately, since the foam is not ideally dry) 120◦ angles. A pentagon is convexand at high pressure, with respect to its neighbours. Picture courtesy of S. Courty [5].

4.3 What Is the Problem?

At first sight, the ideal foam appears truly simple. The expression of its energy, (1), isone of the simplest one can dream of. It immediately yields very intuitive consequences,the above Plateau rules, which considerably reduce the number of degrees of freedomnecessary to describe the structure: for each vertex, an angle, modulo2π/3; for eachside, a length and a curvature.

Now, we give ourselves the area of each of theN bubbles, and the topology of thefoam, that is the list of which bubbles share a common edge, and we wonder: what canwe say about the structure of the foam, the bubble shape, or the total perimeter? Theanswer is that there is nothing simple! The disorder inherent to the foam, especiallydue to the area distribution, but also due to the unavoidable foam boundaries, preventsus from any intuition. There is an intricate interplay between topology, geometry, andpressures, specific to foams, which make them both fascinating and difficult to solve, asthe next sections will discuss.


5 Electrostatic Analogy

5.1 Pressure, Geometry, Topology

One bubble: Consider a bubble of a foam; letn be its number of sides. It is char-acterised by a topological quantity which quantifies the deviation from a hexagonallattice: its topological charge6 − n. Equivalently, we can say that ann-sided bubble isa disinclination bearing a geometrical charge:

q = (6 − n)π

3= 2π − nπ

3. (4)

Now, circle once counterclockwise around this bubble. What the Gauss theoremstates, is here particularly simple: the tangent vectort along the bubble edge, whichtravels around the bubble once counterclockwise, rotates by2π. Notei the number ofthis bubble,j its neighbours. Each sideij is an arc of lengthij and of curvatureκij ,thus contributesκijij to this rotation, while each vertex rotates the tangent vectort byπ/3. The Gauss theorem hence links the average curvature of a bubble to its number ofsides [23]: ∑

j

1γ

(Pi − Pj) ij =∑

j

κijij = (6 − ni)π

3= qi. (5)

This classical but essential equation is what makes 2D foams unique. It relates thetopology, the geometry (curvatures, lengths), and the pressure (hence forces) in a singlebubble. This means for instance that 4-sided or 5-sided bubbles are convex, and theirpressure is higher than their neighbours’; see the hatched pentagon in Fig. 4. On theopposite, 7-sided or 8-sided bubbles are concave and have a low pressure. If this relationwas not satisfied, the bubble would not be at equilibrium: the walls would move, and thepressure adapt, until the condition (5) is fulfilled.

Several bubbles:Extension to more than one bubble uses a topological theorem [30, 2].Consider a contourC which follows only bubble edges and encloses a few bubbles, seethe black contour in Fig. 4. Following the notation of [2], we define the number ofvertices which originate an outward (inward) pointing edge asv+ (v−) (Fig. 4); e.g.v+ = 0 if C is the boundary of the total foam,v− = 0 if C encloses a single bubble.ThetopologicalGauss theorem [30, 2] states thatC encloses a total topological charge∑

k∈C(6 − nk) equal to:

∑k∈C

(6 − nk) = (6 − v+ + v−), (6)

wherek labels the bubbles enclosed byC. For instance, ifC encloses a single bubble,v− = 0, v+ = n, and one recovers (5).

We can write the Gauss theorem, exactly as we wrote it above for one bubble, bystating that the tangent vectort along the contourC, which travels around the bubbleonce counterclockwise, rotates by2π. We thus find that:


Fig. 4. A schematic of a 2D foam. The black contourC follows bubble edges and enclosesa geometrical chargeQ (Q = π/3 due to the pentagon). Vertices with squares point inwards(v− = 5) and vertices with circular discs point outwards (v+ = 10). The dashed contourCcrosses bubble edges transversely. Along the contours tangentt and normaln vectors (arrows)are defined by the right-hand rule. Drawing by Y. Jiang [15].

∑C

κijij = (6 − v+ + v−)π

3. (7)

Combining (6) and (7), we have expressed the geometrical Gauss theorem for aclosed contourC which follows the bubble edgesij and encloses a total geometricalchargeQ [15]:

∑j

Pi − Pj

γij =

∑C

κijij = Q(C) =∑k∈C

qk. (8)


This theorem establishes at a global level the essential relation between topology, ge-ometry and pressure in 2D foams.

Finally, (3) can also be generalised for a contourC crossing the bubble edges per-pendicularly, see the dashed contour in Fig (4). The same argument that applied to acontour around a single vertex now yields, for the whole contour, that the curvatures ofall edges encountered add to zero:

∑C

κ = 0. (9)

5.2 Continuous Limit

We now rewrite the above equations as an analogy with 2D electrostatics (Table 1), anddiscuss its advantages [15].

Table 1. Proposed analogy between foams and electrostatics in two dimensions.

potential field charge

2D potential electric field electricelectrostatics V −∇V chargee

2D pressure curvature geometricalfoams P κ ∝ −∇P charge(6 − n)π

3

In this analogy, each bubble represents a conducting platelet bearing a chargeq =(6−n)π/3 distributed along its edges, which (along each link) is uniform at equilibriumbecause the charge density is proportional to the curvature. The walls are a thin layerof insulating material, which bear a large gradient of pressure (almost a discontinuity).Since we are not interested in the actual pressure within them, we have thus chosento interpolate the pressure in such a way that∆P = 0 within walls. We introduce thenotationE ≡ −∇P :

E ∝ κ n (across walls),E = 0 (within bubbles), (10)

whereκ is the curvature of the walls andn its outwards normal.The pressure thus appears as a potential, the curvature as its gradient (Table 1). Let

the unit vectorsn, t be the normal and the tangent toC, respectively, as drawn in Fig. 4.In the dry foam limit (see [15] for details), (3,8) can be rewritten under a more familiarcontinous form:

198 Franc¸ois Graner∮

C

E · n d ∝ Q(C),∮

C

E · t d = 0. (11)

These approximate relations hold for any contourC (not necessarily parallel or perpen-dicular to edges), although it is easier to visualise (11a) on the black contour, (11b) onthe dashed one. Although both (11a,b) look similar, they are physically different, de-scribing respectively an outwards flux throughC (which, in 2D, is a line integral insteadof a surface integral) and a circulation alongC. They make the analogy to electrostaticsobvious.

5.3 Examples

Pressure and curvature fields: A positiveq corresponds to a local high pressure. Inturn the pressure gradient correlates with the concavity of the bubbles. For illustration,we consider a foam with all bubbles having the same areaA, that is a honeycomb ofsideL. Take one single “defect” bubble with a geometrical chargeq = (6 − n)π/3 asa germ bubble (disinclination), for instance a regular pentagon, hatched in Fig. 4. Thefoam consists of concentric shells around this germ. When the foam is sufficiently large,in thes shell,s � 1, the asymptotic limits of bothκ andP are approximately :

κ(s) ∝ q

Ls,

P = γ∑

s

κ(s) ∝ P0 − γq

Lln s. (12)

HereP0 is a constante.g.equal to the pressurePb at the boundary of the foam, andP grows logarithmically with the foam size. Equations (12) are characteristic of 2Delectrostatics.

Dipoles and quadrupoles:Foams thus enter the well-known class of universality of 2DCoulomb interactions, which encompasses e.g. electrostatics, lines of disinclinations, orarrays of vortices. Two opposite charges−q+ q (e.g.a pentagon-heptagon pair, Fig. 5a)a distanced ∼ L apart constitute a dislocation, that is a dipole of momentp = qd, whichdeforms neighboring bubbles and induces curved edges in the hexagons around it. At adistances � d away from this dipole, the pressure goes approximately as1/s and thecurvature as1/s2: similar to the dipolar potential in 2D. A dipole can pair with anotherdipole (Fig. 5a) to form a topological quadrupole (Fig. 5b), which affects the honeycomblattice over a much shorter range, the pressure going as1/s2 and the curvature as1/s3.

6 The Energy of the Foam

6.1 Perimeter Increase due to a Defect

The analogy is deeper and extends to the expression of energy itself. The energy, or sumof perimeters, of a 2D foam is similar to the energy in 2D electrostatics. We can even


Fig. 5. (a) Simulation of two dipoles in a regular foam with equal areas (δA/ 〈A〉 = 0.4%)and periodic boundaries. Two pentagon-heptagon pairs result in a curvature field in the hexagonsaround them. (b) An artificially constructed pentagon-heptagon-pentagon-heptagon cluster formsa topological quadrupole, with the rest of the honeycomb lattice undisturbed. Simulation by Y.Jiang [15]

define an energy density for 2D foams. The demonstration applies to honeycomb latticeswith only few topological defects, and proceeds as follows.

Energy density: In a honeycomb, when a topological defect induces a curvature inthe neighboring edges, the total foam perimeter increases. To estimate this increase,consider two vertices separated by a distancea. An arc with curvatureκ connectingthem has a length = 2α/κ, whereα = arcsin(aκ/2) is half of the subtended angle.The difference betweena and is the length increase due to the curvature:

− a ≈

[κ22

24+O

(κ44

)]. (13)

As expected, − a is positive and quadratic inκ. This is the expression of the energydensity, proportional to the square of the field.

Energy: By summing over all edges of the foam, and keeping only the leading orderwhenκ � 1, we obtain the increase in energy due to curvature:

Ht ≈ γ∑i<j

ij × κ2ij

2ij

24. (14)

For instance, around the single topological charge,κ(s) decreases approximately asq/s (12). The summation over all curved edges yields the energy cost due to this singletopological defect:


Ht ∝ γ∑

s

sLκ2(s)

∝ γLq2∑

s

1s. (15)

It grows logarithmically with the size of the foam, as expected in analogy with the self-energy of a 2D electrostatic charge. For a large foam, a single topological charge costsso much energy that it never occurs in real foams.

6.2 Several Topological Charges

Consider two chargesq, q′ separated by a (topological) distances � L, the interactionenergy goes approximately as:

Ht ≈ −qq′γL ln( s

L

). (16)

It decreases when charges of same sign separate, or charges of opposite sign aggre-gate. Heptagon-pentagon pairs are more likely than isolated heptagons. This “effectiveinteraction” of topological origin, which previous work has assumed [28] or derivedempirically [25], has important consequences. For instance, this interaction could ex-plain the origin of the correlations between bubbles: the side number distribution andthe Aboav-Weaire law [28]. As an illustration, in Fig. 6, the heptagon (indicated by anumber 7) has two pentagonal neighbors.

This description remains valid even for many charges, as long as the curvature fieldsthey induce are small and can be added by linear superposition. The pressure fields ofmultiple charges are added linearly too. However, the present calculations assume equalareas; they become more difficult when the area disorder couples with the topologicaldisorder.

6.3 Relation Between Pressure and Energy

For a free foam, in any equilibrium state, there exists a relation between the energy andthe pressure field. Surprisingly, it is strictly exact:

H = 2N∑

i=1

(Pi − Pb)Ai, (17)

wherePb is the pressure of the outer fluid at the boundary of the foam. This generalisesin D dimensions:

H =D

D − 1

N∑i=1

(Pi − Pb)Ai. (18)

There exist different demonstrations [1]; the shortest is probably the following, based ona Legendre transformation, usingPb as a Lagrange multiplier [15]. Consider a free foamwith fixed pressures, not areas. At equilibrium, the enthalpy,H ≡ H−∑N

i=1(Pi−Pb)Ai,is extremal. Thus, under dilation,L → λD−1L, A → λDA, the function:


Fig. 6. Annealed 2D ferro-fluid foam.Foam preparation:An ionic magnetic fluid (aqueous mag-netic liquid, black) and oil (white-spirit, grey) do not mix, their surface tension is 15 mN/m. Theyare trapped together in the 1 mm space between two parallel Plexiglas plates. A homogeneousmagnetic field of 9 kA/m perpendicular to the plates induces the cellular structure and fixes thebubble areas and wall thickness. The same oil as that fills the bubbles surrounds them. For detailssee [7].Annealing procedure:We tilt the Plexiglas plates from the horizontal plane to an angleof 0.1◦, inducing a low effective gravity field. Large bubbles drift upwards, small bubbles down-wards, resulting in vertical sorting according to size. We then bring the plates back to horizontal,and the bubbles slowly drift and settle. This procedure allows the bubbles to rearrange and explorethe energy space to find a lower energy configuration. The picture displays the final stable pattern.Picture size is 10 cm. Picture by E. Janiaud, for details see [15].

H(λ) = λD−1H − λDN∑

i=1

(Pi − Pb)Ai, (19)

is extremal atλ = 1; this proves (17,18).


7 The Pressure Field within the Foam

The electrostatic analogy emphasises the central role played by the pressure. How canwe determine experimentally the pressure field within a 2D foam? We summarise andcompare different methods which could be tried.

7.1 Measurements

Direct measurements: An advantage of 2D foams is that each bubble is accessiblefrom outside. It is possible to replace one of the glass plates by an array of pressuresensors [F. Elias, private communication]. This method would present no fundamentaldifficulty, and would yield an unambiguous result, requiring only a compromise betweenthe price of the equipment and the desired precision. The following methods do not useany specific equipment, only image analysis.

Through areas: The measurement of bubble areas yields access to the pressure ifstringent conditions are met. The bubble must be compressible and its equation of stateknown (e.g. a perfect gas). The area variations must result from pressure variations,not from variation of the amount of matter enclosed in the bubble. The pressure of thebubble must be known at least at one time (for instance at equilibrium with atmosphericpressure). Each bubble area must be tracked individually in time. This method has beensuccessfully applied to a 2D soap froth (M. Asipauskas, to be published).

Through coarsening: Conversely, the measurement of bubble area variations appliesunder a different set of conditions. The area variation is due only to diffusion of matterfrom one bubble to its neighbour across their wall, resulting in the coarsening of thefoam. The pressure difference is related with the flow rate across the wall, itself relatedwith the normal velocity of the foam if each bubble is incompressible.

Through curvatures: By measuring the curvatures from a picture, and using iterativelyeqs (2), the pressurePi in each bubble can be determined up to an additive constantP0,the pressure in a reference place (a given bubble, or the outer medium). We only needto chose a path which links bubblei with the reference place, and sum the algebraiccurvatures of the walls crossed. Two different paths would give the same result: thisis readily seen by joining both path to close a circuit, and apply (9) or its continuousequivalent (10b). Thus the choice of the path is arbitrary, and the result is unambiguous.

In practice, this method is imprecise and suited only to very dry foams with goodimage contrast, due to the notorious difficulty of curvature measurements [O. Cardoso,private communication]. In the example of Fig. 6 the error is around 15 % [15]. Animprovement could be to measure all curvatures, typically 3N measurements, and tryto find theN pressures which best fit the 3N (2).


Through vertex positions: M.-L. Chabanol obtained a significant progress by notingthat vertex positions yield as much information as curvatures. In fact, the vertex positionsyield the length of the chord which joins them, which approximates the value of the wall’slength. We thus rewrite the essential (5):

∑j

Pi − Pj

γij = (6 − ni)

π

3.

It is easy to solve theseN linear equations withN unknownPi, i = 1, ..., N . We thenobtain approximate values of thePis, from which we deduce approximateκij andij .Reinjecting these values ofij in theN ×N equations systems enables an iteration untilstable values for pressures, curvatures and lengths are found.

Preliminary tests with C. Monnereau-Pittet and N. Pittet have yielded a quick con-vergence to surprisingly good results. The method is correct even on images with lowcontrast, thick walls or bad focus, because vertex positions and topologies are well mea-sured. The method seems operational, and it is easy to include boundaries. It seemsto converge to a unique set of solutions. Care is required: since the method does notenforce the rule of 120◦ angles, the solution corresponds to a real foam only if thepositions fed in the system are the positions of a foam’s actual vertices. Applicationsare in progress: they could concern extremely precise determinations of curvatures, forinstance for coarsening studies.

7.2 Laplacian of the Pressure

The analogy with electrostatics indicates that anN×N equations system can be broughtback to a single equation. In fact, (10) imply that the Laplacian of the pressureP isproportional to the topological charge. We were thus tempted to conjecture that thetopology, along with boundary conditions, uniquely fixes the pressure field [15]. Thisis not strictly true, since the bubble shapes are not rigidly fixed; in fact, D. Weaire andS. Cox found a counter-example [38]: two patterns, with exactly the same topology andbubble areas, but a different set of pressures, curvatures and lengths.

However, the Laplacian point of view becomes very interesting on larger scales. Ona “mesoscopic” scale much larger than one bubble, but smaller than the foam size, thenet topological charge of the foam is zero [16]. This implies that:

(mesoscopic) ∆P ≈ 0. (20)

The mesoscopic average ofP thus has all properties of zero-laplacian fields. For instance,it is a very “smooth” field; minima or maxima of pressure can exist only on the boundaryof the foam (like Earnshaw theorem in electrostatics [22]). If the foam is free, i.e.surrounded by an outer medium with uniform pressure (Fig. 6), its pressure is thusroughly uniform. If the foam is in a closed box, its maxima are imposed by the shape ofthe box; hence, as N. Pittet noticed, the pressure field and hence the coarsening shoulddiffer in round, square and hexagonal boxes. If the foam is flowing in a channel, thereis a pressure gradient fom the entrance to the exit. This mesoscopic point of view helpsconsidering the foam as a continuous medium [21].


Fig. 7. Artificial side-swapping, or T1 process. We start from an initial equilibrium state (a). Ametallic pin placed above the ferro-fluid foam locally channels the magnetic field lines and attractssome ferro-fluid. This enables us to locally move a vertex and bring it closer to the second one,until they merge and form an unstable 4-fold vertex. The foam spontaneously relaxes towards anew equilibrium state with a different topology (b): bubbles have swapped a side and changedtheir neighbours. Although the energy difference between (a) and (b) was much smaller thanour sensitivity, (b) has visibly more regular bubbles, hence likely a lower energy than (a). Theskelettised image (d) is the difference between (a) and (b), stressing the range of modificationsinduced by this T1: typically 3 bubble sizes. Inducing the reverse T1 was possible (c), and we fallback in the same state, see the comparison between (a) and (c) displayed in (e): both (a) and (b)are lasting equilibrium states. However, the reverse T1 was much more difficult: since the 4-foldvertex systematically decayed into configuration (b), we had to nucleate the side we wanted toappear. Hence the energy barrier between (a) and (b) is asymmetric, again suggesting that (b) hasa lower energy than (a). Picture by F. Elias. For details see [8].


8 The Minimum Perimeter Problem

8.1 Different Equilibrium States

Topological changes:When the foam is at equilibrium, it remains there. However, theequilibrium can be modified, either by mechanical perturbation such as shear, or byvariation of control parameters such as the content of each bubble, through diffusion. Ifone wall length happens to vanish, four bubbles come in contact. Such four-fold vertexis unstable and decays into one of either possible configuration. The foam spontaneouslychoses one, likely the one with the lowest energy, and eventually reaches a new equilib-rium (Fig. 7). If bubbles have exchanged their neighbours, this elementary topologicalchange is called a side-swapping, or T1 process. A T1 process conserves not only thetotal charge but also the total dipolar moment, altering only the quadrupolar moment ofthe foam: it is a current of dipoles.

The energy difference between both equilibrium states is usually much smaller thanthe barrier which separates them (Fig. 7). In fact, creating a four-fold vertex costs oforder the line tension times a wall length, typically 10−7 J. The foams with the lowestbarrier, namely in Langmuir monolayers, have a cost larger than a pN× µm, i.e. atleast 3 orders of magnitude larger thankBT (kBT being reached at the scale of thenm, which is the realm of micro-emulsions rather than foams). Thus a T1 never occursspontaneously under thermal fluctuations, and local energy minima are metastable.

Position of the problem: This raises the question: if we could vary freely the topology,keeping the area distribution fixed, what would be the global energy minimum, i.e. thelowest accessible value of the energy? And what would be the ground state(s), i.e. thecorresponding configuration(s)?

This question, called “the minimal perimeter problem”, is extremely challenging[20]. Mathematicians are progressing at a quick pace, but exact results still mainlyconcern small numbers of bubbles, typicallyN = 2, 3 [18]. However, studying a fewbubbles does not help understanding foams with largeN . Since a topological changecan modify the shape of bubbles over a distance of order three bubble sizes (Fig. 7d),the problem is non-local.

In fact, the mathematical result which is the most useful for physicists, is so usefuland natural that physicists believed it was obvious: if all bubbles have the same areaA, the best pattern is a regular hexagonal lattice! The global energy minimum is then23/231/4λ

√A. This famous “honeycomb conjecture” is now a theorem, recently proved

by Hales [17]. We use this strong result as a starting point for the following approximate,large scale approach [15].

8.2 Global Energy Minimum (Minimal Perimeter at Free Topology)

Anestimate: If bubbles have different areas, not all bubbles need be hexagons. However,it surprisingly happens that regular bubbles obeying Plateau rules, withn curved edgesmeeting at2π/3 angles, all have almost the same ratio of perimeter to square root ofarea: of order23/231/4 ≈ 3.722..., and always larger than2(π)3/2/3 ≈ 3.712... [15].


We thus suggest thatH is close to the energyHh it would have if all bubbles wereregular hexagons:

H ≈ Hh = 3.72 λN∑

i=1

√Ai = 3.72 λN

⟨√A

⟩, (21)

so thatHh is a good approximation of the foam’s global energy minimum. This estimate

depends only on the area distribution of the bubbles. It depends on⟨√

A⟩

, while our

intuition rather often uses〈A〉. If the foam has a boundary, there is an additional termof order

√N 〈A〉.

Tests: We have checked this estimate by examining different examples: simulations andexperiments on disordered foams [15], analytical and numerical calculations on periodicfoams [10], small clusters with large boundary contributions [32]. The main result whichemerges is that the estimate (21) seems consistently correct, its precision being typicallythe percent. The foam energy is usually slightly higher thanHh, but surprisingly, wecould find ordered foams with a value slightly lower thanHh, see Sect. 8.3.

In addition, most configurations tend to have an energy close to the global minimum,and it is very difficult to prepare a foam with an energy a few percent higher (Fig. 8).This explains for instance why we could not measure the energy variation after a T1process (Fig. 7). This means that the energy estimate is an acceptable approximation. It isoperational, since it depends on the area distribution, which is an intrinsic characteristicof the foam, but not on any particular topology.

8.3 Ground State Configuration (Perimeter-Minimizing Pattern)

Difficulty: Since most patterns have almost the same energy, the problem of determiningthe ground state configuration(s) is subtle. Physical foams come always close to the globalenergy minimum, not close to a ground state configuration.

Equation (17) indicates that the energy reaches its minimum when the largest bubbleshave as low a pressure as possible. It is tempting to use it to determine the groundstate configuration. However, this research direction has not yet been successful, to ourknowledge [M. Fortes, private communication]. The reason is probably that the pressurefield is “smooth”, as mentioned above (Sect. 7.2); it might differ from bubble to bubblebut these spatial fluctuations remain local and invisible at large scales, hence difficult toanalyse in details.

We thus use another, approximate approach, to find the ground state configuration(s),in two opposite extreme cases.

Size-sorting: Consider two bubbles inside a foam, with areasA1, A2 respectively. IfA1/A2 is close to 1, the total perimeter should be minimal if both bubbles are close toregular hexagons. Both bubbles share an edge, which length is a compromise between3.722

6

√A1 and3.722

6

√A2, where3.722

6 ≈ 21/2

33/4 ≈ 0.62 is the side of a regular hexagon of


1 1.05 1.1 1.15 1.2 1.25 1.3 1.351.01

1.015

1.02

1.025

1.03

Hi / H

h

Hf /

Hh

Fig. 8. Test of the estimate for the minimum energy. We simulated out-of-equilibrium, highlydeformed foams using the Potts model [15], with a high initial energyHi. The foam spontaneouslyrelaxes towards an equilibrium state, with a final energyHf . Note the difference between verticaland horizontal scales: although we could reach initial energiesHi up to 30% higher thanHh, thefinal energy is systematically the same, barely higher thanHh. Each symbol corresponds to onefoam. Simulations by Y. Jiang [15].

area 1. We have suggested and checked that, in average, the common edge has a lengthof order:

L(A1, A2) ≈ 3.7226

√A

1/21 A

1/22 ≈ 0.62(A1A2)1/4. (22)

Thus the cost in total perimeter is lower when bubbles have neighbours of almost thesame area. This leads to size-sorting, with bubbles segregating according to their sizes.

We have tried to observe this effect by producing an annealed foam, i.e. a foam asclose as possible to the ground state configuration. This is difficult, since the energybarrier between different equilibrium states is larger than the energy differences (Fig. 7).We have succeeded by using ferrofluid-foams, easier to manipulate thanks to their highviscosity / low relaxation time. This size-sorting is favoured by a large area distribution,which is here continuous, so that each bubble can find neighbours of almost the samearea and be as regular as possible. Such size-sorting is visible on Fig. 6.

Size-mixing: On the opposite, when the area distribution is bidisperse, andA1/A2 isvery different from 1, bubbles do not remain 6-sided; they rather change their topology.


10−3

10−2

10−1

100

λ

3.70

3.71

3.72

3.73

3.74

3.752P

* /(1+

λ1/2 )

3191

4282

4181

5272

6161

Fig. 9. The best periodic patterns found by Fortes and Teixeira [10]. Two types of cells, of areasA1 = λ andA2 = 1, are mixed in equal numbers. The vertical axis is twice the perimeter,divided by(1 + λ1/2) to enable comparison withHh (21). For each value ofλ, only the bestpattern found is plotted. The figures indicate the numbers of bubble sides, the indices indicate thenumber of bubbles per unit cell. For instance6161 indicates alternate hexagons;3191 indicatesan array of large bubbles paving a regular honeycomb, but decorated on each other corner by asmall three-sided bubble: surprisingly, this beats the value of regular bubbles,2π3/2/3 (dashedline). Graph by P. Teixeira.

Large bubbles ignore small ones, small bubbles decorate the vertices between large ones.Fortes and Teixeira [10] have calculated the energies of different periodic patterns, andlooked for the most favorable one, for each given value ofA1/A2. Their results appearin Fig. 9. Starting fromA1/A2 = 1, and decreasingA1/A2, they found that the bestpattern is successively: alternate hexagons, then mixture of 5- and 7- sided bubbles, thenmixtures of 4 and 8 (two types of periodic tilings), and finally of 3 and 9.

Recently, we have found [31] that the periodic tiling of 9-sided and 3-sided bubbles(large bubbles placed in honeycomb lattice, with tiny 3-sided bubbles decorating half oftheir vertices) beatsHh, and even slightly beats the value 3.712λ

∑Ni=1

√Ai (Fig. 9).

We are currently investigating this surprising result.


Fig. 10. A rather exotic 2D foam! This is a true experiment, actually a ferro-fluid foam. Doyou think the 0-angle (i.e. round) bubble can be at equilibrium? And the 1-angle bubble? Picturecourtesy of C. Flament.

9 Conclusion

We have tried here to show how foams link mathematical properties, such as topologyand curvature, with physical quantities such as energy and pressure (Sect. 5.1). We haveadopted the point of view of statistical physics, linking the apparently simple propertiesof each bubble (Sect. 4) with the global behaviour of a disordered foam.

We have considered here an ideal 2D foam (Sect. 3.2) at equilibrium, where eachbubble has a given constant area. We have tried to show the central role of the energyand pressure field. We can estimate both on large scales without knowing microscopicdetails. The energy is approximately a function of the area distribution only, and more

precisely on⟨√

A⟩

(21); it is almost equal to the perimeter it would have if each bubble

had a regular shape, and can even be slightly lower (Sect. 8.3). The pressure resemblesa field with zero laplacian (Sect. 7.2), as we could understand using an analogy with 2Delectrostatics (Sect. 5). Both quantities are also related together (Sect. 6.3).

We currently try to open perspectives into three directions: actual 3D foams; mechan-ical properties of foams out of equilibrium; and more complex fluids such as emulsionsand elastic-plastic-viscous materials. We hope to have shown the triple interest of foamresearch: as a truly fascinating interdisciplinary field, as a carefully-chosen model sys-tem, and for their own applications.


Acknowledgments

The results presented here have been derived together with M. Aubouy, S. Courty, F.Elias, J.A. Glazier, Y. Jiang. I would also like to acknowledge the collaboration withM. Asipauskas, O. Cardoso, M.-L. Chabanol, S.J. Cox, R. Delannay, B. Dollet, C.Flament, M. Fortes, S. Ifadir, E. Janiaud, O. Lordereau, C. Monnereau-Pittet, N. Pittet,P. Teixeira, M.F. Vaz. Thanks are also due to G. Debregeas, K. Kassner, N. Kern, J.Lajzerowicz, K. Mecke, F. Morgan, D. Weaire for useful discussions. D. Weaire andF. Elias critically read the manuscript. L. Mousson, from the Association des Buveursd’Orges (www.space.ch/abo), was an endless source of information about beer foams.

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